We present high-resolution 870-um ALMA continuum maps of 30 bright sub-millimeter sources in the UKIDSS UDS field. These sources are selected from deep, 1-square degrees 850-um maps from the SCUBA--2 Cosmology Legacy Survey, and are representative of the brightest sources in the field (median SCUBA2 flux S_850=8.7+/-0.4 mJy). We detect 52 sub-millimeter galaxies (SMGs) at >4-sigma significance in our 30 ALMA maps. In 61+/-17% of the ALMA maps the single-dish source comprises a blend of >=2 SMGs, where the secondary SMGs are Ultra--Luminous Infrared Galaxies (ULIRGs) with L_IR>10^12 Lo. The brightest SMG contributes on average 80+/-4% of the single-dish flux density, and in the ALMA maps containing >=2 SMGs the secondary SMG contributes 25+/-3% of the integrated ALMA flux. We construct source counts and show that multiplicity boosts the apparent single-dish cumulative counts by 20% at S_870>7.5mJy, and by 60% at S_870>12mJy. We combine our sample with previous ALMA studies of fainter SMGs and show that the counts are well-described by a double power-law with a break at 8.5+/-0.6mJy. The break corresponds to a luminosity of ~6x10^12Lsol or a star-formation rate of ~1000Mo/yr. For the typical sizes of these SMGs, which are resolved in our ALMA data with r=1.2+/-0.1kpc, this yields a limiting SFR density of ~100Msol/yr/kpc2. Finally, the number density of S_870>2mJy SMGs is 80+/-30 times higher than that derived from blank-field counts. An over-abundance of faint SMGs is inconsistent with line-of-sight projections dominating multiplicity in the brightest SMGs, and indicates that a significant proportion of these high-redshift ULIRGs must be physically associated. ; Comment: 14 pages, 6 figures. ApJ in press